Theorem uspgrushgr 29508

Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)

Assertion

Ref	Expression
uspgrushgr	⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Proof of Theorem uspgrushgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29483	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		ssrab2 4035	. . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		f1ss 6783	. . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	4, 5	mpan2 703	. . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	3, 6	biimtrdi 256	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	1, 2	isushgr 29392	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
9	7, 8	sylibrd 262	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph))
10	9	pm2.43i 53	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Syntax hints:   → wi 4   ∈ wcel 2143  {crab 3416   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590   class class class wbr 5110  dom cdm 5663  –1-1→wf1 6535  ‘cfv 6538   ≤ cle 11245  2c2 12296  ♯chash 14368  Vtxcvtx 29327  iEdgciedg 29328  USHGraphcushgr 29388  USPGraphcuspgr 29479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546  df-ushgr 29390  df-uspgr 29481

This theorem is referenced by:  uspgrupgrushgr  29510  usgredgedg  29561  vtxdusgrfvedg  29822  1loopgrvd2  29834  isubgr3stgr  48723